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ANSWER: 5-4 Sum and Difference Identities Find the exact value of each trigonometric expression. 4. cos SOLUTION: SOLUTION: Write 6. tan as the sum or difference of angle Write as the sum or difference of angle measures with cosines that you know. measures with cosines that you know. ANSWER: 6. tan SOLUTION: Write as the sum or difference of angle measures with cosines that you know. ANSWER: 2− 10. TECHNOLOGY A blind mobility aid uses the same idea as a bat’s sonar to enable people who are visually impaired to detect objects around them. The sound wave emitted by the device for a certain patient can be modeled by b = 30(sin 195 )t, where t is time in seconds and b is air pressure in pascals. a. Rewrite the formula in terms of the difference of two angle measures. b. What is the pressure after 1 second? SOLUTION: a. b. eSolutions Manual - Powered by Cognero Page 1 The pressure after 1 second is –7.8 pascals. ANSWER: ANSWER: a. Sample answer: b = b. about –7.8 pascals 5-4 Sum and Difference Identities 2− Find the exact value of each expression. cos – cos sin 10. TECHNOLOGY A blind mobility aid uses the same idea as a bat’s sonar to enable people who are visually impaired to detect objects around them. The sound wave emitted by the device for a certain patient can be modeled by b = 30(sin 195 )t, where t is time in seconds and b is air pressure in pascals. a. Rewrite the formula in terms of the difference of two angle measures. b. What is the pressure after 1 second? 14. sin SOLUTION: SOLUTION: a. ANSWER: b. 15. cos 40 cos 20 – sin 40 sin 20 SOLUTION: ANSWER: The pressure after 1 second is –7.8 pascals. 16. ANSWER: SOLUTION: a. Sample answer: b = b. about –7.8 pascals Find the exact value of each expression. cos – cos sin 14. sin SOLUTION: ANSWER: Simplify each expression. 17. SOLUTION: eSolutions Manual - Powered by Cognero Page 2 ANSWER: ANSWER: ANSWER: ANSWER: 5-4 Sum and Difference Identities Simplify each expression. 17. sin 4y Write each trigonometric expression as an algebraic expression. 24. sin (arcsin x + arccos x) SOLUTION: SOLUTION: ANSWER: If we let and , then . Sketch one right triangle and with an acute angle and another with an acute angle . Label the sides such that and . Then use the Pythagorean Theorem to express the length of each third side. tan θ 18. cos cos x + sin sin x SOLUTION: ANSWER: sin x 19. sin 3y cos y + cos 3y sin y Using these triangles, we find that SOLUTION: and values in ANSWER: . Substitute these and simplify. sin 4y Write each trigonometric expression as an algebraic expression. 24. sin (arcsin x + arccos x) SOLUTION: ANSWER: If we let 1 and , then . Sketch one right triangle and with an acute angle and another with an acute angle . Label the sides such that and . Then use the Pythagorean Theorem to express the length of each third side. 26. SOLUTION: eSolutions Manual - Powered by Cognero Page 3 ANSWER: 26. 5-4 Sum and Difference Identities SOLUTION: Verify each cofunction identity using one or more difference identities. 32. tan = cot x SOLUTION: If we let and and , then . Sketch one right triangle with an acute angle and another with an acute angle . Label the sides such that and . Then use the Pythagorean Theorem to ANSWER: express the length of each third side. Using these triangles, we find that , , and . Substitute these values in and 33. sec = csc x simplify. SOLUTION: ANSWER: ANSWER: Verify each cofunction identity using one or more difference identities. eSolutions Manual - Powered by Cognero Page 4 5-4 Sum and Difference Identities 33. sec = csc x 38. sin (90 + ) = cos SOLUTION: SOLUTION: ANSWER: Find the solution to each expression on the interval [0, 2π). ANSWER: 40. cos – sin = 0 SOLUTION: when x = On the interval [0, 2 ), and x = Verify each reduction identity. + ) = cos 36. cos (2 ANSWER: , SOLUTION: 42. cos ANSWER: 38. sin (90 + . + sin = 0 SOLUTION: ) = cos SOLUTION: = ANSWER: when x = On the interval [0, 2 ), and x . ANSWER: , eSolutions Manual - Powered by Cognero Page 5 43. sin Find the solution to each expression on the + sin = = . ANSWER: 5-4 Sum and Difference Identities , 43. sin + sin = SOLUTION: On the interval [0, 2 ), = when x = and x . ANSWER: , eSolutions Manual - Powered by Cognero Page 6